Bellman Expectation Equation leading to results where value iteration would not converge to the optimal policy
What am I missing here?

You are not missing anything mathematically.
Potentially what you are missing is that the discount factor $\gamma$, is part of the problem definition. In reinforcement learning (RL), you do not always solve problems to obtain the highest total sum of rewards. Instead you solve problems to obtain the highest expected return on any action. If you use a discount factor, then it affects the return calculation, which in turn may affect which behaviours are optimal.
If you choose $gamma$ such that an agent would move away from a terminal state to receive the highest expected return, then you have defined the problem such that this behaviour is optimal.
This can be an issue when solving RL problems and you have taken a free choice for reward functions and definition of return (this is quite common, often it is the researcher's problem to set these values up to define the agent's goals). Often you can use a discount factor to make a problem more numerically stable, because it will keep sums of rewards over many time steps within bounds. Also, it can be used as a way to search for faster solutions - i.e. in less time steps to reach episode end - because the agent will prioritise reaching higher values sooner if it can. Your example MDP deliberately puts those two effects in opposition to each other.
Assuming you want the agent to find the terminal state by getting over the larger negative reward just before it: Probably you should not use a discount factor for your MDP. Alternatively, if the reward signal is more of a free choice for that problem, you could offset it (by $+0.1$) and then most values of discount factor will still resolve to reaching the terminal state as optimal behaviour.
There are a few common ways to mitigate or avoid the issue of changing problem definition by selecting $\gamma$:

Use a value of $\gamma$ derived from the problem. In some cases, the value of $\gamma$ is a realistic physical part of a problem. For instance in financial problems, cash now is often preferable to cash later.

Set $\gamma$ to relavtively high value, such as $0.99$ or $0.999$, depending on episode length and relative sizes of rewards. This can work well enough for a wide range of problems. It is a common solution when using DQN. Effectively this approximates the next approach (average rewards) for a low cost.

Use average reward settings. There are separate Bellman equations, TD updates etc based on maximising expected average reward per time step as opposed to maximising expected discounted return.